Hybrid aggregation operator and its application to multiple attribute decision making problems

By combining the advantages of the additive weighted mean (AWM) operator and the ordered weighted averaging (OWA) operator, this paper first presents a hybrid operator for aggregating data information, and then proposes a hybrid aggregation (HA) operator-based method for multiple attribute decision making (MADM) problems. The theoretical analyses and the numerical results show that the HA operator generalizes both the AWM and OWA operators, and reflects the importance of both the given argument and the ordered position of the argument. Thus, the HA operator can reflect better real situations in practical applications. Finally, an illustrative example is given.

Multiple attribute decision making method based on trapezoid fuzzy linguistic variables

The problem of multiple attribute decision making under fuzzy linguistic environments, in which decision makers can only provide their preferences （attribute values）in the form of trapezoid fuzzy linguistic variables（TFLV）, is studied. The formula of the degree of possibility between two TFLVs is defined, and some of its characteristics are studied. Based on the degree of possibility of fuzzy linguistic variables, an approach to ranking the decision alternatives in multiple attribute decision making with TFLV is developed. The trapezoid fuzzy linguistic weighted averaging （TFLWA） operator method is utilized to aggregate the decision information, and then all the alternatives are ranked by comparing the degree of possibility of TFLV. The method can carry out linguistic computation processes easily without loss of linguistic information, and thus makes the decision results reasonable and effective. Finally, the implementation process of the proposed method is illustrated and analyzed by a practical example.

Multiple attribute decision making based on different types of linguistic information

Distance measures between exact linguistic variables and between uncertain linguistic variables are introduced respectively. Based on exact linguistic variables and uncertain linguistic variables, the concepts of positive linguistic ideal solution and negative linguistic ideal solution of attribute values are defined. To rank and select alternatives, based on the distance measures of two types of linguistic variables and the linguistic ideal solutions, a method for multiple attribute decision making with different types of linguistic information is proposed, by which all alternatives can be ranked. The method can carry out linguistic computation processes easily without loss of linguistic information, and thus makes the decision result reasonable and effective. Finally, the implementation process of the proposed method is illustrated and analyzed by a numerical example.

New approach to multiple attribute decision making with interval numbers

In an ambiguous decision domain, the evaluation values of alternatives against attributes would be interval numbers because of the inherent, uncertain property of the problems. By using a number of linear programming models, Bryson and Mobolurin propose an approach to compute attribute weights and overall values of the alternatives in the form of interval numbers. The intervals of the overall values of alternatives are then transformed into points or crisp values for comparisons among the alternatives. However, the attribute weights are different because of the use of linear programming models in Bryson and Mobolurin＇s approach. Thus, the alternatives are not comparable because different attribute weights are employed to calculate the overall values of the alternatives. A new approach is proposed to overcome the drawbacks of Bryson and Mobolurin＇s approach. By transforming the decision matrix with intervals into the one with crisp values, a new linear programming model is proposed, to calculate the attribute weights for conducting alternative ranking.

Entropy-based procedures for intuitionistic fuzzy multiple attribute decision making

The class of multiple attribute decision making （MADM） problems is studied, where the attribute values are intuitionistic fuzzy numbers, and the information about attribute weights is completely unknown. A score function is first used to calculate the score of each attribute value and a score matrix is constructed, and then it is transformed into a normalized score matrix. Based on the normalized score matrix, an entropy-based procedure is proposed to derive attribute weights. Furthermore, the additive weighted averaging operator is utilized to fuse all the normalized scores into the overall scores of alternatives, by which the ranking of all the given alternatives is obtained. This paper is concluded by extending the above results to interval-valued intuitionistic fuzzy set theory, and an illustrative example is also provided.

Method for multiple attribute decision making based on incomplete linguistic judgment matrix

With respect to the multiple attribute decision making problems with linguistic preference relations on alternatives in the form of incomplete linguistic judgment matrix, a method is proposed to analyze the decision problem. The incomplete linguistic judgment matrix is transformed into incomplete fuzzy judgment matrix and an optimization model is developed on the basis of incomplete fuzzy judgment matrix provided by the decision maker and the decision matrix to determine attribute weights by Lagrange multiplier method. Then the overall values of all alternatives are calculated to rank them. A numerical example is given to illustrate the feasibility and practicality of the proposed method.

Maximizing deviation method for multiple attribute decision making under q-rung orthopair fuzzy environment

Because of the uncertainty and subjectivity of decision makers in the complex decision-making environment,the evaluation information of alternatives given by decision makers is often fuzzy and uncertain.As a generalization of intuitionistic fuzzy set(IFSs)and Pythagoras fuzzy set(PFSs),q-rung orthopair fuzzy set(q-ROFS)is more suitable for expressing fuzzy and uncertain information.But,in actual multiple attribute decision making(MADM)problems,the weights of DMs and attributes are always completely unknown or partly known,to date,the maximizing deviation method is a good tool to deal with such issues.Thus,combine the q-ROFS and conventional maximizing deviation method,we will study the maximizing deviation method under q-ROFSs and q-RIVOFSs in this paper.Firstly,we briefly introduce the basic concept of q-rung orthopair fuzzy sets(q-ROFSs)and q-rung interval-valued orthopair fuzzy sets(q-RIVOFSs).Then,combine the maximizing deviation method with q-rung orthopair fuzzy information,we establish two new decision making models.On this basis,the proposed models are applied to MADM problems with q-rung orthopair fuzzy information.Compared with existing methods,the effectiveness and superiority of the new model are analyzed.This method can effectively solve the MADM problem whose decision information is represented by q-rung orthopair fuzzy numbers(q-ROFNs)and whose attributes are incomplete.

Solution to multiple attribute group decision making problems with two decision makers

A kind of multiple attribute group decision making （MAGDM） problem is discussed from the perspective of statistic decision-making. Firstly, on the basis of the stability theory, a new idea is proposed to solve this kind of problem. Secondly, a con- crete method corresponding to this kind of problem is proposed. The main tool of our research is the technique o~ the jackknife method. The main advantage of the new method is that it can identify and determine the reliability degree of the existed decision making information. Finally, a traffic engineering example is given to show the effectiveness of the new method.

Power Aggregation Operators and Similarity Measures Based on Improved Intuitionistic Hesitant Fuzzy Sets and their Applications to Multiple Attribute Decision Making

Intuitionistic hesitant fuzzy set(IHFS)is amixture of two separated notions called intuitionistic fuzzy set(IFS)and hesitant fuzzy set(HFS),as an important technique to cope with uncertain and awkward information in realistic decision issues.IHFS contains the grades of truth and falsity in the formof the subset of the unit interval.The notion of IHFS was defined by many scholars with different conditions,which contain several weaknesses.Here,keeping in view the problems of already defined IHFSs,we will define IHFS in another way so that it becomes compatible with other existing notions.To examine the interrelationship between any numbers of IHFSs,we combined the notions of power averaging(PA)operators and power geometric(PG)operators with IHFSs to present the idea of intuitionistic hesitant fuzzy PA(IHFPA)operators,intuitionistic hesitant fuzzy PG(IHFPG)operators,intuitionistic hesitant fuzzy power weighted average(IHFPWA)operators,intuitionistic hesitant fuzzy power ordered weighted average(IHFPOWA)operators,intuitionistic hesitant fuzzy power ordered weighted geometric(IHFPOWG)operators,intuitionistic hesitant fuzzy power hybrid average(IHFPHA)operators,intuitionistic hesitant fuzzy power hybrid geometric(IHFPHG)operators and examined as well their fundamental properties.Some special cases of the explored work are also discovered.Additionally,the similarity measures based on IHFSs are presented and their advantages are discussed along examples.Furthermore,we initiated a new approach to multiple attribute decision making(MADM)problem applying suggested operators and a mathematical model is solved to develop an approach and to establish its common sense and adequacy.Advantages,comparative analysis,and graphical representation of the presented work are elaborated to show the reliability and effectiveness of the presented works.

A Method for Risky Multiple Attribute Decision Making with Four - dimensional Reference Point

A method based on cumulative prospect theory was proposed to solve risky multiple attribute decision making problems with Four -dimensional reference points. Considering the influence of different learning processes and corresponding features on decision-making, a new reference-learning behavior is added, and a risk-based multiple-attribute decision-making method based on four-dimensional reference point cumulative prospect theory is proposed. Firstly, according to the cumulative prospect theory, the prospect value and the decision function value of the four reference points of learning, time, evaluation value and expected value are calculated respectively, and the cumulative prospect value matrix of each program dynamic is formed. Secondly,according to the WAA operalor, Maximize the stage weighting model to obtain the integrated cumulative prospect value. Finally, on the basis of this, the alternatives are sorted according to the size of the total cumulative prospect value, and compared with other methods, the validity and scientific of the proposed method are proved.

Variable precision rough set for multiple decision attribute analysis

A variable precision rough set (VPRS) model is used to solve the multi-attribute decision analysis (MADA) problem with multiple conflicting decision attributes and multiple condition attributes. By introducing confidence measures and a β-reduct, the VPRS model can rationally solve the conflicting decision analysis problem with multiple decision attributes and multiple condition attributes. For illustration, a medical diagnosis example is utilized to show the feasibility of the VPRS model in solving the MADA problem with multiple decision attributes and multiple condition attributes. Empirical results show that the decision rule with the highest confidence measures will be used as the final decision rules in the MADA problem with multiple conflicting decision attributes and multiple condition attributes if there are some conflicts among decision rules resulting from multiple decision attributes. The confidence-measure-based VPRS model can effectively solve the conflicts of decision rules from multiple decision attributes and thus a class of MADA problem with multiple conflicting decision attributes and multiple condition attributes are solved.

Maclaurin Symmetric Mean Aggregation Operators and Their Application to Hesitant Q-Rung Orthopair Fuzzy Multiple Attribute Decision Making

The Maclaurin symmetric mean(MSM)operator exhibits a desirable characteristic by effectively capturing the correlations among multiple input parameters,and it serves as an extension of certain existing aggregation operators through adjustments to the parameter k.The hesitant q-rung orthopair set(Hq-ROFSs)can serve as an extension of the existing orthopair fuzzy sets,which provides decision makers more freedom in describing their true opinions.The objective of this paper is to present an MSM operator to aggregate hesitant q-rung orthopair numbers and solve the multiple attribute decision making(MADM)problems in which the attribute values take the form of hesitant q-rung orthopair fuzzy sets(H-qROFSs).Firstly,the definition of H-qROFSs and some operational laws of H-qROFSs are proposed.Then we develop a family of hesitant q-rung orthopair fuzzy maclaurin symmetric mean aggregation operators,such as the hesitant q-rung orthopair fuzzy maclaurin symmetric mean(Hq-ROFMSM)operator,the hesitant q-rung orthopair fuzzy weighted maclaurin symmetric mean(Hq-ROFWMSM)operator,the hesitant q-rung orthopair fuzzy dual maclaurin symmetric mean(Hq-ROFDMSM)operator,the hesitant q-rung orthopair fuzzy weighted dual maclaurin symmetric mean(Hq-ROFWDMSM)operator.And the properties and special cases of these proposed operators are studied.Furthermore,an approach based on the Hq-ROFWMSM operator is proposed for multiple attribute decision making problems under hesitant q-rung orthopair fuzzy environment.Finally,a numerical example and comparative analysis is given to illustrate the application of the proposed approach.

Man-machine Function Allocation Based on Uncertain Linguistic Multiple Attribute Decision Making

Function allocation is one of the necessary stages in the design course of man-machine systems since appropriate function allocation makes the whole system more effective, reliable and inexpensive. Therefore, our research mainly focuses on the problems of function allocation between man and machine in man-machine systems, analyses each capability advantage of man and machine according to their respective inherent characteristics and makes a comparison between them. In view of highly uncertain characteristics of decision attribute value in the practical process, we introduce the uncertain linguistic multiple attribute decision making （ULMADM） method in the function allocation process. Meanwhile, we also use the uncertain extended weighted arithmetic averaging （UEWAA） method to determine the automation level range of the operator functions. Then, we eventually estab- lish the automation level of man-machine function allocation by using the multi-attribute decision making algorithm, which is combined by UEWAA and uncertain linguistic hybrid aggregation （ULHA） operators. Finally, an example about function allocation is given, that is, fault diagnosis in the cockpit of civil aircraft. The final result of the example demonstrates that the proposed method about function allocation is feasible and effective.

Fuzzy Multiple Attribute Decision Making for Evaluating Aggregate Risk in Green Manufacturing

Industrial risk and the diversification of risk types both increase with industrial development. Many uncertain factors and high risk are inherent in the implementation of new green manufacturing methods. Because of the shortage of successful examples and complete and certain knowledge, decision-making methods using probabilities to represent risk, which need many examples, cannot be used to evaluate risk in the implementation of green manufacturing projects. Therefore, a fuzzy multiple attribute decision-making （FMADM） method was developed with a three-level hierarchical decision-making model to evaluate the aggregate risk for green manufacturing projects. A case study shows that the hierarchical decision-making model of the aggregate risk and the FMADM method effectively reflect the characteristics of the risk in green manufacturing projects.

Interval Neutrosophic Prioritized OWA Operator and Its Application to Multiple Attribute Decision Making

On the basis of prioritized aggregated operator and prioritized ordered weighted average(POWA)operator,in this paper,the authors further present interval neutrosophic prioritized ordered weighted aggregation(INPOWA)operator with respect to interval neutrosophic numbers(INNs).Firstly,the definition,operational laws,characteristics,expectation and comparative method of INNs are introduced.Then,the INPOWA operator is developed,and some properties of the operator are analyzed.Furthermore,based on the INPOWA operator and the comparative formula of the INNs,an approach to decision making with INNs is established.Finally,an illustrative example is given to verify the developed approach and to demonstrate its practicality and effectiveness.

Expected Value Method for Fuzzy Multiple Attribute Decision Making

This paper presents a fuzzy multiple attribute decision-making （FMADM） method in which the attribute weights and decision matrix elements （attribute values） are fuzzy variables. Fuzzy arithmetic and the expected value operator of fuzzy variables are used to develop the expected value method to solve the FMADM problem. A numerical example is given to demonstrate the feasibility and effectiveness of the method.

Novel method for hybrid multiple attribute decision making based on TODIM method

The TODIM(an acronym in Portuguese for interactive and multiple attribute decision making) method is a valuable tool to solve the multiple attribute decision making(MADM) problems considering the behavior of the decision maker(DM), while it cannot be used to handle the problem with unknown weight information on attributes. In this paper, a novel method based on the classical TODIM method is proposed to solve the hybrid MADM problems with unknown weight information on attributes,in which attribute values are represented in four different formats:crisp numbers, interval numbers, triangular fuzzy numbers and trapezoidal fuzzy numbers. Firstly, the positive-ideal alternative and negative-ideal alternative are determined, and the gain and loss matrices are constructed by calculating the gain and loss of each alternative relatived to the ideal alternatives concerning each attribute based on different distance calculation formulas, which may avoid the information missing or information distortion in the process of unifying multiform attribute values into a certain representation form. Secondly, an optimization model based on the maximizing deviation(MD) method, by which the attribute weights can be determined, is established for the TODIM method. Further, the calculation steps to solve the hybrid MADM problems are given. Finally, two numerical examples are presented to illustrate the usefulness of the proposed method, and the results show that the DM's psychological behavior, attribute weights and the transformed information would highly affect the ranking orders of alternatives.

Hesitant Trapezoid Fuzzy Hamacher Aggregation Operators and Their Application to Multiple Attribute Decision Making

In this paper,we investigate the multiple attribute decision making(MADM)problems in which the attribute values take the form of hesitant trapezoid fuzzy information.Firstly,inspired by the idea of hesitant fuzzy sets and trapezoid fuzzy numbers,the definition of hesitant trapezoid fuzzy set and some operational laws of hesitant trapezoid fuzzy elements are proposed.Then some hesitant trapezoid fuzzy aggregation operators based on Hamacher operation are developed,such as the hesitant trapezoid fuzzy Hamacher weighted average(HTr FHWA)operator,the hesitant trapezoid fuzzy Hamacher weighted geometric(HTr FHWG)operator,the hesitant trapezoid fuzzy Hamacher Choquet average(HTr FHCA),the hesitant trapezoid fuzzy Hamacher Choquet geometric(HTr FHCG),etc.Furthermore,an approach based on the hesitant trapezoid fuzzy Hamacher weighted average(HTr FHWA)operator and the hesitant trapezoid fuzzy Hamacher weighted geometric(HTr FHWG)operator is proposed for MADM problems under hesitant trapezoid fuzzy environment.Finally,a numerical example for supplier selection is given to illustrate the application of the proposed approach.

Linear goal programming approach to obtaining the weights of intuitionistic fuzzy ordered weighted averaging operator

The multiple attribute decision making problems are studied, in which the information about attribute weights is partly known and the attribute values take the form of intuitionistic fuzzy numbers. The operational laws of intuitionistic fuzzy numbers are introduced, and the score function and accuracy function are presented to compare the intuitionistic fuzzy numbers. The intuitionistic fuzzy ordered weighted averaging （IFOWA） operator which is an extension of the well-known ordered weighted averaging （OWA） operator is investigated to aggregate the intuitionistic fuzzy information. In order to determine the weights of intuitionistic fuzzy ordered weighted averaging operator, a linear goal programming procedure is proposed for learning the weights from data. Finally, an example is illustrated to verify the effectiveness and practicability of the developed method.

UNIFORMIZATION OF MULTIGRANULAR LINGUISTIC LABELS AND THEIR APPLICATION TO GROUP DECISION MAKING

In multiple attribute group decision making （MAGDM） problems based on linguistic information, the granularities of linguistic label sets are usually different due to the differences of thinking modes and habits among decision makers. In order to deal with this inconvenience, the transformation relationships among multigranular linguistic labels （TRMLLs）, which are applied to unify linguistic labels with different granularities into a certain linguistic label set with fixed granularity, are presented in this paper. Furthermore, the reference tables are made according to TRMLLs so that the interrelated calculation will be less complicated, and the method of how to use them is explained in detail. At length, the TRMLLs are illustrated through an application example.